Simplicity is not Simple:
Tessellations and Modular Architecture
Euclidean plane.
There are many kinds of tessellations, from the
one given by simple square bathroom tiles to tessellations involving lizards or demons in the artwork
of M. C. Escher to the non-periodic Penrose tilings
of the plane. The simplest kinds of planar tessellations are those like the bathroom tiles where the subregions are identical, regular, convex polygons. In
these tessellations, all angles and sides are identical.
Architecturally, this means that if you were to use
such a tessellation to construct office workstations,
you would only need one basic type of panel and one
basic type of connector. The resulting cubicles would
all be equal sized, and arranged in a regular pattern.
The 2000 MathFest in Los Angeles was an extravaganza of mathematical talks, short courses and exhibits. By the third day, we needed a break from
math and decided to check out the city. Of course,
after being immersed in mathematics for so long, we
noticed it everywhere. In particular, while walking
down a street in a remote neighborhood, we happened
upon a rhombic dodecahedron in a shop window. It
was a hanging lamp, in a gallery full of fantastical
hip furniture. We went in, fully expecting that our
shabby clothing would result in a cool reception (we
later discovered that the chairs sold for $3000 apiece).
Much to our surprise, we were shown enthusiastically
around the shop by a man who turned out to be the
designer, Gregg Fleishman. He took us into his workshop, and showed us how he had been using rhombic
dodecahedra to develop a modular building system.
Despite his claims that he was not very good at math,
it became clear as we talked that mathematics was
the foundation for much of his work.
Planar tessellations and Floor Plans
The fact that squares and rectangles tessellate the
plane explains why they are used so often in architecture. Using squares or rectangles, floor plans can
be constructed that have no wasted space between
rooms. Of the regular polygons (convex polygons
where all sides have equal length and all angles are
equal) there are only three that tessellate the plane:
triangles, squares and hexagons (Figure 1).
In this article, we’ll introduce you to Fleishman’s
work, modular architecture more generally, and talk
about how various architectural considerations can
be described in mathematical terms. Along the way,
we’ll discuss and prove some basic facts about polyhedra and tessellations.
What is Modular Architecture?
Modular Architecture is any building system in which
a few standardized components used to build a structure on a scale much larger than the components. Ideally, the parts should be easy to duplicate and simple
to assemble by one person. To get a sense of what is
meant by this, consider first the mathematically simpler problem faced by a company that wants to break
up a large workroom into smaller identical workstations. Generally, rather than having a construction
crew come in and build walls, the company will buy
a set of identical dividers and connectors which fit
together to create the cubicles. They buy as many
as they need to complete their project, and the parts
can be assembled without specialized crews.
Figure 1: The three regular tessellations.
It is not hard to prove that these are the only three
tessellating regular polygons. Th following proof is
taken from Keith Critchlow’s book Order in Space,
The Viking Press, 1970.
Proof. Let P be a regular polygon with n > 2 sides.
Then each interior angle of P measures α = 180− 360
n
degrees. If P tessellates the plane then each vertex of
the tessellation will be surrounded by k(n) = 360
α =
4
2 + n−2
polygons. It now suffices to determine which
integers n > 2 will make k(n) an integer. If n > 7
4
< 1 and thus k(n) could not be an
then 0 < n−2
integer. Checking the remaining possible values for n
we see that k(3) = 6, k(4) = 4, k(5) = 2 43 , and k(6) =
3. Thus only 3-, 4-, and 6-sided regular polygons can
tessellate the plane.
Mathematically, the problem involved is one of
breaking up a planar region into subregions using a
finite number of identical pieces. These subregions
should fit together so that they cover the whole plane
without overlapping, except on edges. Such a method
of subdivision is called a tiling or a tessellation of the
1
Although squares are more often used than hexagons in architecture today, the hexagon has a mathematical advantage over both the square and the triangle. A hexagonal tessellation has the most efficient
perimeter-to-area ratio, and thus will require the least
amount of wall material per square foot of floor space.
high school with identical hexagonal classrooms that
tile very efficiently, but hallways and larger rooms
like cafeterias or libraries could be difficult to place
in such a floor plan (see William Blackwell’s text Geometry in Architecture, John Wiley & Sons, 1984).
Another problem is that so much architecture is already rectangular that other shapes often don’t fit as
Proof. A square with side length E has area E 2 and well in the available spaces.
perimeter
√ 4E, so the perimeter of a square with area
A is 4 A. An
(equilateral) triangle with side length Modular Buildings
√
3 2
E has area 4 E , so a triangle with area A will have
√
Office cubicle systems are a two-dimensional version
3√
perimeter 3E = 2(3) 4 A ≈ 4.559 A. √
Finally, a of modular building systems. Gregg Fleishman is inhexagon with side length E has area A = 3 2 3 E 2 and terested in creating entire buildings, not just floor
√
1
1√
thus perimeter 6E = 2(2) 2 (3) 4 A ≈ 3.722 A.
plans within existing buildings. This means he is
interested in ways of subdividing three-dimensional
For example, a square room enclosing 100 square space. Designing a house, for instance, means creatfeet requires a perimeter of 40 feet. A triangular room ing an enclosed space and separating off rooms within
with the same square footage would need a bit more that possibly multi-floored structure. The goal is still
perimeter, about 45.59 feet. The most efficient floor to do this with only a few basic parts.
unit is the hexagon, which requires a perimeter of
Consider how you could build a backyard clubonly about 37.22 feet to enclose 100 square feet of house. If you had five identical square pieces of plyarea.
wood, you could build a simple cube-shaped clubA workspace which is closer to a circle in shape
also wastes less space in inaccessible corners. An office worker sitting in a swivel chair in the middle of
an office can reach everything within a circular area
whose radius is that worker’s reach. A worker in a
hexagonal office can more easily reach everything in
the workspace than a worker in a square or rectangular cubicle. Further, hexagonal workstations can
be arranged to fill circular or odd-shaped spaces. For
all of these reasons, some companies are turning to
hexagonal cubicle systems (Figure 2). The Silicon
Valley company Nokia used hexagonal cubicles from
Herman Miller, Inc. to fit eleven workstations and a
couch area in a circular space where only six to eight
rectangular workstations could have fit. Employees
feel that they have more room and a more flowing
office space with this hexagonal system, according to
the report Resolve connects with Nokia on the Herman Miller, Inc., website at www.hermanmiller.com/
us/pdfs/resolve/nokia cs.pdf, 2000.
house with a dirt floor. Rectangular prisms (“boxes”)
are the most common basic solid used in architecture.
Cubes are the simplest of them in that all the faces
are equivalent. Hence even a couple of grade school
kids can build one. A solid where every face is the
same regular polygon and all vertices (corners) are
alike is called a regular or Platonic solid. There are
exactly five of these: the tetrahedron, the cube, the
octahedron, the dodecahedron and the icosahedron.
This can be proved in a few ways. Perhaps the prettiest uses a number called the Euler characteristic.
Proof. Given a surface, S, a decomposition of S is a
way of cutting S up into pieces so that each piece can
be flattened into a planar region with no holes. The
pieces are called the faces of the decomposition, the
lines along which the faces were cut apart are called
the edges, and the corners where edges connect are
called the vertices.
The Euler characteristic of S, denoted χ(S), is
then given by the formula χ(S) = F − E + V , where
F is the number of faces, E is the number of edges,
and V is the number of vertices in any decomposition
of the surface (see figure ??). It is a theorem that this
number does not depend on how the surface is decomposed, and also doesn’t change if you stretch or bend
the surface (the Euler Characteristic is thus an example of a topological invariant). For a proof of this,
see pp. 29–32 of Algebraic Topology: An Introduction, W.S. Massey, Graduate Texts in Mathematics
** missing figure **
Figure 2: A hexagonal office cubicle system by Herman Miller, Inc.
Despite the mathematical efficiency of hexagons,
in real life they are often not as convenient as square
or rectangular floor shapes. One problem is the difficulty in placing hallways and larger classrooms in
hexagonal systems. For example, one could build a
2
56, Springer-Verlag 1967.
for cubicles. An ideal structure would enclose a large
Every Platonic solid gives a decomposition of the amount of space with a small amount of building masphere, which you can see by imagining inflating the terial.
solid until its faces bulge out to a sphere. Using
“Another issue is the shape of the space. If you
the tetrahedron, for example, we get a decomposi- use other shapes, like the rhombic dodecahedron, you
tion with four faces, twelve edges and eight vertices, can get a higher and more interesting ceiling and roof.
so we know that χ(sphere)= 4 − 12 + 8 = 2. So we Finally, the closer a building is to spherical, the better
know that for any Platonic solid, we have the equa- it is able to withstand high winds and other weather.”
tion F − E + V = 2.
These types of considerations led modular buildNow we can use the defining properties of Platonic
solids to show there can only be five of them. Since
the faces of a Platonic solid must be regular n-sided
polygons for some n ≥ 3, we know that the total
number of face-edge intersections must be n times
the number of faces, since each face meets n edges.
Also, each edge meets two faces, so we get a second
equation nF = 2E. Similarly, since each vertex is
regular, each vertex meets the same number of edges,
call it m, while each edge meets two vertices. Thus
we get a third equation coming from counting the
number of edge-vertex intersections, mV = 2E. Each
vertex must meet at least 3 edges, so m is also at least
3.
ing pioneer Buckminster Fuller to start experimenting with geodesic domes in the 1940’s. “Fuller generally started with the icosahedron and then subdivided each face into smaller triangles,” explained
Fleishman. “Of course, the object that results is
no longer a Platonic solid, and can be difficult to
build. You have to use a great number of rods of various lengths that intersect at various angles. Building
large domes requires using a computer to keep track
of where the different length rods go.” Domes (large
ones) have nevertheless been popular. Two of the
best known examples are the Epcot Center at Disneyworld and the NSF Amundsen-Scott South Pole Station (to see pictures of these architectural domes, visit
Solving for F and V in terms of E in the sec- www.disneyworld.go.com/ waltdisneyworld/parksanond and third equations and plugging into the first dmore/attractions and www.stanford.edu/group/zarelab/antartica.slide21.html).
equation, we obtain:
Fleishman’s personal goals have led him to experiment with different types of modular building systems. “One of my goals has always been to deal with
a way of building the house, and I spent a long time
on dome design. Geodesic dome architecture has to
do with building a frame, then covering it, but covering the frame is frequently difficult. Also, a sphere
is one object, and it is difficult to connect to others
like it, so to make a large house, the only choice is to
make a large sphere, and then you still have all of the
interior walls to construct. In residential structures I
have found it makes sense to go to a panel structure.”
Thus Fleishman has been working on creating a system of modular panels which can attach to create a
variety of buildings with several joined smaller rooms
rather than a single large dome.
2E
2E
−E+
= 2.
n
m
Rearranging this, we get:
1
1
2
1
+
=
+ .
n m
E
2
This tells us that 1/n + 1/m > 1/2. Since n and m
must each be at least 3, this gives us five possibilities.
The first, n=3 and m=3, gives a solid in which three
(since m=3) equilateral triangles (since n=3) meet at
each vertex, that is, the tetrahedron. When n=3 and
m=4, we get the octahedron, n=3 and m=5, gives
the icosahedron, n=4 and m=3 gives the cube, and
finally, n=5 and m=3, gives the dodecahedron.
If cubes and rectangular prisms are so easy to
build with and so common in architecture, what is the
motivation for exploring different systems? We asked
Mr. Fleishman about this, who explained, “Cubes
are useful, they are part of the solution I have arrived at. I am not against the cube at all. But it
has drawbacks. First, cubes have a low volume to
surface-area ratio. Of all three-dimensional shapes,
the sphere has the highest. The closer you get to approximating a sphere, the better structural economy
you have.” This is akin to the perimeter to area issue
Where does this leave us mathematically? Well,
we are still hoping to create a structure using only
a few different types of simple pieces, so we want
to use Platonic solids if possible, or at least shapes
with some of the same regularity properties, like being built of regular polygons, or having all faces the
same. In addition, we now want to be able to connect
rooms to each other in a space-filling way. So what we
need is a three-dimensional version of a tessellation
involving regular or semi-regular solids.
3
Space-filling solids
This leads us back to cubes and rectangular prisms.
There is a reason besides simplicity that they are the
most common solid used in architecture: they tessellate 3-space. That is, a collection of rectangular
prisms of the same size can be arranged to fill 3dimensional space so that there no gaps in between.
Although these are the most common architectural
building blocks, they are not the only solids that fill
space. However, they are the only regular polyhedron
which does (top of Figure 3). Thus to find other polyhedra that tessellate 3-space, we need to relax the
regularity conditions. There are a couple of ways to
do this. Either you can relax the condition that the
faces all be the same regular polygons, or you can Figure 3: The only regular polyhedron that tesselrequire the faces to be identical, but allow them not lates 3-space is the cube. The only Archimedean
to be regular polygons.
polyhedron that fills space is the truncated ocThe first possibility leads us to Archimedean solids: tahedron (bottom left).
The only space-filling
polyhedra whose faces are regular polygons (possibly Archimedean dual is the rhombic dodecahedron (botof more than one variety) where every vertex is identi- tom right).
cal. There are thirteen Archimedean polyhedra, and
of the thirteen the only one that fills 3-space is the
the volume formula will be some
truncated octahedron (bottom left of Figure 3). See edge length, and
3
multiple
of
E
.
Thus
the function writing surface
Alan Holden’s book Shapes, Space, and Symmetry,
area
in
terms
of
volume
will be of the form SA(V ) =
Columbia University Press, 1971.
2
αV 3 for some constant α. Table 1 lists the volume
If we replace the vertices of an Archimedean poly- and surface areas for the cube, rhombic dodecahedron
hedron by faces and replace the faces by vertices we (R.D.) and truncated octahedron (T.O.) in terms of
get the Archimedean dual of the polyhedron. This their edge length E, and an approximate value for
gives us the second possibility; the faces will be iden- the associated coefficient α.
tical, but they may not be regular polygons. Of
the thirteen Archimedean duals, only one tessellates
three space: the rhombic dodecahedron (bottom right Table 1: Volume and surface area formulae for three
of Figure 3). See Order in Space, Keith Critchlow, space-filling solids. Smaller values of α indicate more
The Viking Press, 1970.
efficient space-filling.
In general, there are only eight space-filling polyhedra, three of which are described above. Of these
eight, the truncated octahedron (a polyhedron with
six square faces and eight hexagonal faces) fills space
most efficiently, i.e. has the most volume for the least
surface area. The rhombic dodecahedron (a twelvesided polyhedron whose faces are diamonds) is the
second most efficient. The cube ranks a measly sixth
out of eight in surface area to volume efficiency (rankings can be found in Critchlow’s Order in Space, The
Viking Press, 1980). By manipulating surface area
and volume formulas we can prove that the truncated
octahedra is slightly more efficient than the rhombic
dodecahedron and significantly more efficient than
the cube.
Solid
Cube
R.D.
T.O.
Volume
E
3
√
16
9√
3E
8 2 E3
Surface Area
2
3
6E
√
8 2 E2
√
(6 + 12 3) E 2
≈α
6.0000
5.3454
5.3147
For example, a cube-shaped room enclosing 1000
cubic feet would have an edge length of 10 cubic feet
2
and a surface area of αV 3 = 6(100) = 600 square
feet. To enclose the same volume with a rhombic dodecahedron we would only need about 534.54 square
feet of surface area (since α = 5.3454). The most efficient space-filler, the truncated octahedron, can fill
1000 cubic feet of volume with a surface area of only
Proof. The surface area formula for each of these 531.47 square feet. Note that the rhombic dodecasolids will be some multiple of the square E 2 of the hedron is nearly as efficient as the truncated octahedron, but the cube is quite inefficient in comparison.
4
Further Considerations
involved in the rhombic dodecahedron system is very
Fleishman has developed two modular building sys- simple. It is amazing what you can do with simple
tems: one based on the rhombic dodecahedron and math.”
one based on a semi-regular tiling using rhombicuboctahedrons, cubes and tetrahedrons. “The rhombic Housing for the World
dodecahedron is a nice shape because it involves stan- We had a great time playing with Fleishman’s smalldard angles and lengths. The diagonals
of the faces scale building models, and told him we thought they
√
have lengths in the ratio of 1 : 2, the same as the would make great math toys, and he said he’d be
ratio of the length of the side of a square to its di- willing to make some if there was interest. But when
agonal. The dihedral angles (angles between faces) we asked him how he envisioned his modular buildare 120 degrees. This makes it easier for lay people ings being used, he had a much more revolutionary
to machine the parts and build the structures.” Com- plan. “The plywood panel house is lacking insulation
pare this to the truncated octahedron, whose dihedral and waterproofing, but it is very easy to assemble
angles are approximately 125 degrees 16 minutes and and has a lot of potential for variety. My idea is that
109 degrees 29 minutes. Because the dihedral angles it has the perfect form for creating low-cost housing
of the rhombic dodecahedron are all equal and be- in emerging economies. The panel structures could
cause each face has an even number of edges, Fleish- be erected by crews with minimal training to provide
man was able to develop a tab and slot connection interior structure. Exterior surfacing and finishing,
system. Since the panels themselves interlock in this like insulation, could be done using indigenous matesystem (see picture), it eliminates need for a separate rials and techniques. These structures could provide
connector system.
a global solution to the housing problem.”
The system involving the tetrahedron, cube, and
rhombicuboctahedron is based on the rhombic dodecahedron system. This tiling of space is achieved
by truncating the vertices of the rhombic dodecahedrons. We asked Fleishman how these systems compare to each other. “The advantages of this system
over the rhombic dodecahedron system are that instead of the large diamond panel, which most likely
needs to be cast or built up in some manner, there
are square and rectangular panels, which are easier to
cut from stock material. The variety of shapes and
ability to rescale the components also permits a wider
variety of module, or, basically, room sizes. However,
the system also has the disadvantage of having more
component parts and not creating as stable an exterior skin as the rhombic dodecahedron system.” At
the cost of simplicity, Fleishman has also designed
a three way panel connection system for both geometries so that interior walls could be built with
the same panels as the walls. Fleishman never stops
searching for better solutions; the conversations we
had with him while writing this article led him to
reconsider the truncated octahedron as a basic unit.
He is now working on a new system based on that
shape.
For Further Reading
Pictures of Gregg Fleishman’s work and the history of the development of his modular building systems can be found at www.greggfleishman.com. For
more information about polyhedra, see the following
websites: www.physics.orst.edu/bulatov/polyhedra,
www.ics.uci.edu/eppstein/junkyard/tiling.html, and
polytopes.wolfram.com.
Fleishman wants people to be able to build with
his panels without extensive training. “Making a
good system from only a few pieces which are not
too hard to machine or build is quite difficult,” says
Fleishman, “Simplicity is not simple.” “In the end, in
spite of its complex appearnce which has perhaps prevented others from exploring its potential, the math
5